Wednesday, August 29, 2007

What does a mathematical proof show? In what sense, if any, do the objects which mathematicians reason about exist? A program of study on the philosophy of mathematics might consider some of the following views:

Bourbaki - Structuralism

Brower - Intuitionism

Frege - Logicism (?)

Gödel - Platonism

Hilbert - Formalism

Lakatos - Proofs and refutations in the tradition of Popper

J. S. Mill - Math as empirical generalization

Poincare - Intuitionism (?)

Russell - Logicism

Wittgenstein - Constructivism (?)

And one might look at more recent academic interpretations and commentary, such as Putnam or Kripke's interpretation of Wittgenstein.

Philosophers of math often discuss various interesting bits of mathematics. These include the construction of real numbers as equivalence classes of Cauchy-convergent sequences of rationals, of rational numbers as equivalence classes of ordered pairs of integers, and of integers as functions mapping (subsets of) the natural numbers to the natural numbers. Each construction comes with definitions of Those who think mathematics is in need of a foundation have often looked for one in terms of logic and set theory. Different axiom systems have been offered for sets. Russell's theory of types contrasts with the Zermelo-Fraenkel (ZF) system. In these set theories, there is an infinity of orders of infinity. I've always like the proof that the power set of a set, that is, the set of all subsets of a set, cannot be put in a one-to-one relationship with the original set. Thinking about applying that theorem recursively to the set of the natural numbers soon exhausts my imagination. Someday I would like to understand Gödel's proofs that if ZF is consistent, then ZP with the axiom of choice (ZFC) is consistent. And if ZFC is consistent, then Cantor's continuum hypothesis is consistent with ZFC. Paul Cohen went further. He proved, in 1963, the continuum hypothesis is independent of the axioms of ZFC. I guess this relates to model theory. I gather the Löwenheim-Skolem theorem is a surprising result.

Philosophers of mathematics often discuss certain important results from comutability theory and the theory of automata. Among these are Gödel's imcompleteness theorem. (Barkley Rosser, Sr., generalized Gödel’s work, from ω-consistency to consistency.) I gather the unsolvability of Diophantine equations, in general, follows from Gödel's theorem. The existence of uncomputable functions is of interest. Every computer programmer should be aware of the halting problem. I find interesting the Church-Turing thesis, the Chomsky hierarchy, and the question of whether the set of problems that can be solved in polynomial time by a deterministic Turing machine is equivalent to the set of problems that can be solved in polynomial time by a nondeterministic Turing machine.

Monday, August 27, 2007

"Professor Harcourt reviews a body of doctrine challenging the foundations of standard economics. The challenges, if valid, would confer increased respectability on ways of organizing production and income distribution guided more by politics and oriented less toward profit than the ways most amenable to standard analysis." -- Leland B. Yeager (1976). "Review of Some Cambridge Controversies in the Theory of Capital", American Political Science Review, V. 70, N. 4 (December): 1270-1271

Mainstream economists responded to the Cambridge Capital Controversy by claiming that disaggregated models of intertemporal and temporary equilibrium are unaffected. So the question arises whether Cambridge capital-theory “paradoxes” can tell us anything interesting about such models. It is known that multiple equilibria can arise in overlapping generations, single-good, models. So pointing out multiple equilibria is not enough. Fratini addresses this issue. He constructs a reswitching example in which reswitching is necessary for multiple equilibriua to arise at an interest rate in a region where a higher interest rate is associated with more savings.

Here’s the abstract of Fratini’s paper:

"We study an overlapping generation model of Diamond's type. In contrast to Diamond, we do not assume that the technology can be represented by an aggregate neoclassical production function. Rather, we study the case in which (i) the technology consists of a finite number of constant coefficient productive activities, (ii) there are capital goods physically heterogeneous with respect to each other and to the economy's only consumption good and (iii) there are two alternative production methods for obtaining the consumption good. We explore the possible linkage between reswitching of techniques and the multiplicity of stationary state equilibria of the model."

Fratini’s research is quite close to some of my own research. Let me compare and contrast:

Fratini considers a structure of production in which in each technique, the single consumption good is produced with a different capital good. This approach resembles this example I take from Garegnani, except in Garegnani’s example a continuum of capital goods is available. In Fratini’s model, only two techniques are available.

Fratini imposes some constraints on his model that I do not consider. For example, roughly, he assumes that in each technique the production of the consumption good is more capital-intensive than the production of the capital good for that technique.

Fratini is better on the economic intutition. For example, he explains his savings schedule as combining income and well-behaved substitution effects.

I have described a wider range of overlapping generation models. In particular, I wonder if my needing to include a labor-leisure trade-off in the agents’ utility function (as compared to Section 3 here) to get multiple equilibria associated with the “perverse” switch is a challenge to Fratini’s results. (Keep in mind that I do not impose any restriction on the direction of price Wicksell effects.)

Fratini briefly analyzes the stability of stationary states, drawing on past results in the economic literature.

I think I am better on visualization of results. This is probably only because I was willing to do more tedious work within a spreadsheet that is needed to generate Fratini’s Figures 1 and 2 and Table 1.

Fratini has gone through the labor of writing up his results in a coherent whole and seeing the article through the refereeing and publishing process.

Tuesday, August 21, 2007

Alex Tabarrok praises Phelps desire to develop a non-equilibrium, non-hydralic economics. “GVV”, a commenter, points out that Nicholas Kaldor and Joan Robinson have already developed this theme. This led to a question of whether Austrians or Post Keynesians were first to develop this theme. (Peter Boettke also responds to Phelps and comments by pointing to contributions of the Austrian school.)

I think both schools developed a receptiveness to this theme in parallel. Hayek’s student Shackle emphasized the uncertaintly and money in Keynes. Another of Hayek’s students, Ludwig Lachmann, was important, along with Kirzner, in the revival of interest in the U.S. in the 1970s in the Austrian school. And Lachmann lauded Shackle’s interest in disequibrium.

My point in this post is to point out that these themes of disequilibrum, uncertainty, and historical time were in Keynes’ revolution from the start. Consider:

”Actually, however, we have as a rule, only the vaguest idea of any but the most direct consequences of our acts. Sometimes we are not much concerned with their remoter consequences, even though time and chance may make much of them. But sometimes we are intensely concerned with them, more so, occasionally, than with the immediate consequences. Now of all human activities which are affected by this remoter preoccupation, it happens that one of the most important is economic in character, namely, wealth. The whole object of the accumulatin of wealth is to produce results, or potential results, at a comparatively distant, and sometimes at an indefinitely distant, date. Thus the fact that our knowledge of the future is fluctuating, vague and uncertain, renders wealth a peculiarly unsuitable subject for the methods of the [neo]classical economic theory. This theory might work very well in a world in which economic goods were necessarily consumed within a short interval of their being produced. But it requires, I suggest, considerable amendment if it is to be applied to a world in which the accumulation of wealth for an indefinitely postponed future is an important factor; and the greater the proportionate part played by such wealth-accumulation the more essential does such amendment become.

By ‘uncertain’ knowledge, let me explain, I do not mean merely to distinguish what is known for certain from what is only probable. The game of roulette is not subject, in this sense, to uncertainty; nor is the prospect of a Victory Bond being drawn. Or, again, the expectation of life is only slightly uncertain. Even the weather is only moderately uncertain. The sense in which I am using the term is that in which the prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence, or the obsolence of a new invention, or the position of private wealth-owners in the social system in 1970. About these matters there is no scientific basis on which to form any capable probability whatever. We simply do not know. Nevertheless, the necessity for action and for decision compels us as practical men to do our best to overlook this awkward fact and to behave exactly as we should if we had behind us a good Benhamite calculation of a series of prospective advantages and disadvantages, each multiplied by its appropriate probability, waiting to be summed.

How do we manage in such circumstances to behave in a manner which saves our faces as rational economic men? We have devised for the purpose a variety of techniques, of which much the most important three are:

1. We assume that the present is a much more serviceable guide to the future than a candid examination of past experience would show it to have been hitherto. In other words we largely ignore the prospect of future changes about the actual character of which we know nothing.

2. We assume that the existing state of opinion as expressed in prices and the character of existing output is based on a correct summing up of future prospects, so that we can accept it as such unless and until something new and relevant comes into the picture.

3. Knowing that our own individual judgement is worthless, we endeavor to fall back on the judgement of the rest of the world, which is perhaps better informed. That is, we endeavor to conform with the behavior of the majority or the average. The psychology of a society of individuals each of whom is endeavoring to copy the others leads to what we may strictly term a conventional judgement.

Now a practical theory of the future based on these three principles has certain marked characteristics. In particular, based on so flimsy a foundation, it is subject to sudden and violent changes. The practice of calmness and immobility, of certainty and security, suddenly breaks down. New fears and hopes will, without warning, take charge of human conduct. The forces of disillusion may suddenly impose a new conventional basis of valuation. All these pretty, polite techniques, made for a well-panelled board room and a nicely regulated market, are liable to collapse. At all times the vague panic fears and equally vague and unreasoned hopes are not really lulled and lie but a little way below the surface.

Perhaps the reader feels that this general philosophical disquisition on the behavior of mankind is somewhat remote from the economic theory under discussion. But I think not. Though this is how we behave in the market-place, the theory we devise in the study of how we behave should not itself submit to market-place idols. I accuse the [neo]classical economic theory of being one of those pretty, polite techniques which tries to deal with the present by abstracting from the fact that we know very little about the future.” -- John Maynard Keynes (1937).

Monday, August 20, 2007

The following figure is from Jeffrey M. Stonecash's "The Income Gap" (Political Science and Politics, July 2006):

Voting Differences Greater When Class Divisions Rise

The upper line is the Gini coefficient, multiplied by 100 to fall on a scale ranging from zero to 100. The higher the Gini coefficient, the moe unequal income is distributed. Notice how it has an upward trend after 1972. The income distribution has been becoming more unequal in the United States for a generation.

The difference is, roughly, the difference between the percentage of the poorest 15% voting Democratic and the percentage of the richest 5% voting Democratic. A higher value here shows that the rich and poor have more different voting patterns.

With the exceptions of 2000 and 2004, the electorate seems to increasingly perceive the Democratic party as for the less affluent as income becomes more unevenly distributed. If you want to live like a Republican, vote Democratic. (Stonecash explains the break in 2000 and 2004 on exceptional circumstances - Clinton's impeachment and Iraq. I wonder if it may have something to do with increasing media concentration and dishonesty. An overwhelming proportion of the mass media in the U.S. is owned by a handful of companies.)

Friday, August 17, 2007

Do mainstream economists have a theory of prices? General Equilibrium theory is obviously one candidate for such a theory. But economic theorists have found many problems with it, some internally generated. After examining the failure of General Equilibrium Theory, in a paper in the Cambridge Journal of Economics, Rizvi turned to look at game theory, another candidate:

"It is important to understand the reasons why general equilibrium theory had to be replaced. This has to do with the emergence of arbitrariness results in general equilibrium theory during the 1970s and early 1980s. Because of these results, it became obvious that individual rationality postulates gave no structure to aggregate theorizing: the aggregate realm was undetermined and arbitrary. This vacuum provided an environment in which rational choice game theory, with all of its problems, was able to prosper. Here, too, the conception of rationality was inadequate to the task at hand. Nash play required entirely unbelievable assumptions, and created a plethora of equilibria; the simplest form of subgame perfection was bedeviled by experimental nonconfirmation. These problems gave impetus in turn to the development of boundedly rational and evolutionary approaches to game theory. To the extent that these approaches have had success in giving basis to some notions from rational choice game theory, they do this by, to a large extent, doing away with rationality. Yet since many of these results were obtained in an oversimple setting, it does not seem likely that evolutionary approaches will find widespread economic applications. Another response to the arbitrariness results has been the work on market demand. Explicitly, authors in this tradition do away with individual rationality to the extent possible. What remains, then, of optimization and its relation to best choice? Since best choice has been the basis of normative economics, evaluative concerns are sacrificed in the effort to obtain aggregate-level results of the most straightforward sort. Many of these approaches are to some extent allied to the emergence of experimental economics. But there is controversy concerning the role of experiments in economics. Are they just a source of paradoxes to be resolved in the usual way in theoretical economics, or should experiments form the centerpiece of an inductive approach to economics? If the latter way is found to be attractive, it is doubtful that there is much chance that the view of economic behavior or satisfaction that emerges will be anything like the universal and individually rational agent.

In the course of examining these transformations in economic theory, we can see quite clearly a loss of ambition in terms of generality, and an increasingly tenuous grasp on the concept of rationality. In consequence, there is a loss in the ability to say anything much about normative considerations. To some, the current period of rapid theoretical change and even to some extent pluralism will be attractive; but durable results are hard to find and many of the fundamental concerns of economics have fallen by the wayside." -- S. Abu Turab Rizvi (1999?). "Rationality, Evolution and Games"

Wednesday, August 15, 2007

In keeping with my amateur status, I like the support of references. Other bloggers have different styles.

I think I'll even provide references for an Eric Nilsson post, which I'm almost sure that he is aware of.

The term "Fordism" was used by Antonio Gramsci in his Prison Notebooks. As I understand it, the phrase is also used by those developing the theory of economic regulation and describing social structures of accumulation. A standard reference here, on my list of books to read some time, is Michael Aglietta's A Theory of Capitalist Regulation: The U.S. Experience (Verso, 1979, first published in French in 1976). Lots of work has been done here since 1976.

Monday, August 13, 2007

Over on Crooked Timber, "reason" says he likes a James Galbraith quotation I refer to. Galbraith mentions Koopmans. I suppose Galbraith is talking about this:

"Here the 'direct' implications of the postulates, their accuracy in describing directly observed individual behavior, are placed [by Friedman] in a category with which we need to be less concerned.

There are several objections to such a concept of theory construction. In the first place, in order that we shall have a refutable theory at all, the postulates then need to be supplemented by a clear description of the class of implications by which the theory stands or falls. Otherwise, every contradiction between an implication and an observation could be met by reclassifying the implication as a 'direct' one.

This objection is met by Friedman's suggestion that there should in each case be 'rules for using the model,' that is, a specification of the 'class of phenomena the hypothesis is designed to explain.' But a second objection arises out of this answer to the first. To state a set of postulates, and then to exempt a subclass of their implications from verification is a curiously roundabout way of specifying the content of a theory that is regarded as open to empirical refutation. It leaves one without an understanding of the reasons for the exemptions. The impression of ingeniousness that this procedure gives is reinforced by the fact that in each of Professor Friedman's examples he knows more about the phenomenon in question than he lets on in his suggested postulates. He is willing to predict the expert billard player's shots from the hypothesis that the player knows the mathematical formulae of mechanics and computes their application to each situation with lightning speed, even though he (Friedman) knows that most experts at billards do not have these abilities. He is willing to predict the distribution of leaves on a tree from the hypothesis that each leaf seeks a position of maximum exposure to sunlight (given the position of all other leaves), although no one has reported observing a leaf change its location on a tree.

One cannot help but feel uneasy in the face of so much ingenuity. Truth, like peace, is indivisible. It cannot be compartmentalized. Before we can accept the view that obvious discrepancies between behavior postulates and directly observed behavior do not affect the predictive power of specified implications of the postulates, we need to understand the reason why these discrepancies do not matter. This is all the more important in a field such as economics where, as Friedman also emphasizes, the opportunities for verification of the predictions and implications derived from the postulates are scarce and the outcome of such verification often remains somewhat uncertain..." -- Tjalling C. Koopmans (1957). Three Essays on the State of Economic Science, McGraw-Hill: 139-140

Koopmans also discusses methodology, in an attack on American institutionalism, in his "Measurement without Theory" (Review of Economic Statistics, V. 29, N. 3 (August 1947)). I have already pointed out some more recent criticisms of Friedman's methodology.

Sunday, August 12, 2007

"Equilibrium in production, like equilibrium in exchange, is an ideal and not a real state. It never happens in the real world that the selling price of any given product is absolutely equal to the cost of the productive services that enter into that product, or that the effective demand and supply of services or products are absolutely equal..." -- Walras (1954: Lesson 18, Section 188).

and again:

"Finally, in order to come still more closely to reality, we must drop the hypothesis of an annual market period and adopt in its place the hypothesis of a continuous market. Thus, we pass from the static to the dynamic state. For this purpose, we shall now suppose that the annual production and consumption, which we had hitherto represented as a constant magnitude for every moment of the year under consideration, change from instant to instant along with the basic data of the problem... Every hour, nay, every minute, portions of these different classes of circulating capital are disappearing and reappearing. Personal capital, capital goods proper and money also disappear and reappear, in a similar manner, but much more slowly. Only landed capital escapes this process of renewal. Such is the continuous market, which is perpetuating tending towards equilibrium without ever actually attaining it, because the market has no other way of approaching equilibrium except by groping, and, before the goal is reached, it has to renew its efforts and start over again, all the basic data of the problem, e.g. the initial quantities possessed, the utilities of goods and services, the technical coefficients, the excess of income over consumption, the working capital requirements, etc., having changed in the meantime. Viewed in this way, the market is like a lake agitated by the wind, where the water is incessantly seeking its level without ever reaching it. But whereas there are days when the surface of a lake is almost smooth, there never is a day when the effective demand for products and services equals their effective supply and when the selling price of products equals the cost of the productive services used in making them. The diversion of productive services from enterprises that are losing money to profitable enterprises takes place in various ways, the most important being through credit operations, but at best these ways are slow. It can happen and frequently does happen in the real world, that under some circumstantces a selling price will remain for long periods of time above the cost of production and continue to rise in spite of increases in output, while under other circumstances, a fall in price, following upon this rise, will suddenly bring the selling price below cost of production and force entrepreneurs to reverse their production policies. For, just as a lake is, at times, stirred to its very depths by a storm, so also the market is sometimes thrown into violent confusion by crises, which are sudden and general disturbances of equilibrium. The more we know of the ideal conditions of equilibrium, the better we shall be able to control or prevent these crises." -- Walras (1954: Lesson 35, Section 322).

For Walras, equilibrium is a property of his theoretical model, never of the economy. His claim is that an equilibrium model can tell us something about actual economies because of tendencies existing in the economy at any time.

Now Walras could have misunderstood and mischaracterized his own theory. Perhaps properties of neoclassical theory prevent it from being applied with the method Walras describes. As I havepreviouslyblogged, Sraffians, such as Bharadwaj, Garegnani, Gram, and Petri, have made this case for neo-Walrasian models adopted and developed by economists more recent than Walras. I take the Arrow-Debreu model of intertemporal equilibrium as the canonical example. This is a short-period model, and represents a dramatic change in economic method. In this model, the initial quantites of all capital goods are taken as data. Yet, some of these quantites, that is, those of circulating capital goods, change in any approach to equilibrium. Thus, if an economy is approaching an equilibrium, that equilibrium will not be the Arrow-Debreu equilibrium calculated for the data, as currently existing in the economy.

For the Arrow-Debreu model to be descriptive of existing capitalist economies, such economies must never be out of equilibrium. This is a defect of the theory.

Pierangelo Garegnani (1976). “On a Change in the Notion of Equilibrium in Recent Work on Value and Distribution”, reprinted in Keynes’s Economics and the Theory of Value and Distribution (edited by J. L. Eatwell and M. Milgate, 1983), Oxford University Press

Harvey Gram (1995). “The Role of Perfect Foresight in Krishna Bharadwaj’s Critque of Demand and Supply Equilibrium-Based Theory”, in The Classical Tradition in Economic Thought: Perspectives on the History of Economic Thought: Volume Eleven (edited by I. H. Rima), Edward Elgar

Saturday, August 11, 2007

Paul Krugman's New York Timescolumn yesterday seems to be describing Hyman Minsky's financial instability hypothesis. The obviousness of the practical relevance of Minsky's hypothesis today is indeed a scary thing.

Monday, August 06, 2007

"It is not true that individuals possess a prescriptive 'natural liberty' in their economic activities. There is no 'compact' conferring perpetual rights on those who Have or on those who Acquire. The world is not so governed from above that private and social interest always coincide. It is not so managed here below that in practice they coincide. It is not a correct deduction from the Principles of Economics that enlightened self-interest always operates in the public interest. Nor is it true that self-interest generally is enlightened; more often individuals acting separately to promote their own ends are too ignorant or too weak to attain even these. Experience does not show that individuals, when they make up a social unit, are always less clear-sighted than when they act separately." -- John Maynard Keynes (1926). "The End of Laissez-Faire"

I was inspired to contextualize this Keynes quotation, with the post title, by a reaction on Ezra Klein's blog to Dani Rodrik's dualism. (I like the Keynes quotation that Rodrik chooses.) There are otherreactions to Rodrik, including by me in comments.

Update:Atrios alludes to the Lipsey and Lancaster (1956-1957) paper; his title is an allusion to a funny Leijonhufvud paper.

Update II: Does Radek describe Econ 101 as a kind of "noble lie"? Anyways he has some links to others discussion Rodrik's post. He doesn't link to Peter Boettke, though.

Friday, August 03, 2007

"You say, 'I don't see how demand can be said to have no influence on ... prices, unless constant returns ...' I take it that the drama is enacted on Marshall's stage where the claimants for influence are utility and cost of production. Now utility has made little progress (since the 1870ies) towards acquiring a tangible existence and survives in textbooks at the purely subjective level. On the other hand, cost of production has successfully survived Marshall's attempt to reduce it to an equally evanescent nature under the name of 'disutility', and is still kicking in the form of hours of llabour, tons of raw materials, etc. This, rather than the relative slope of the two curves, is why it seems to me that the 'influence of the two things on price is not comparable." -- Piero Sraffa (1971). Letter to Athanasios Asimakopulos, in (1)

"I am sorry to have kept your MS so long - and with so little result. The fact is that your opening sentence is for me an obstacle which I am unable to get over. You write: 'It is a basic proposition of the Sraffa theory that prices are determined exclusively by the physical requirements of production and the social wage-profit division with consumers demand playing a purely passive role.' Never have I said this: certainly not in the two places to which you refer in your note 2. Nothing, in my view, could be more suicidal than to make such a statement. You are asking me to put my head on the block so that the first fool who comes along can cut it off neatly. Whatever you do, please do not represent me as saying such a thing." -- Piero Sraffa (1964). Letter to Arun Bose, in (2)

Thursday, August 02, 2007

"Finally, there was an even more interesting third assumption implicit and explicit in the classical mind. It was a belief in unique long-run equilibrium independent of initial conditions. I shall call it the 'ergodic hypothesis' by analogy to the use of this term in statistical mechanics. Remember that the classical economists were fatalists (a synonym for 'believers in equilibrium'!). Harriet Martineau, who made fairy tales out of economics (unlike modern economists who make economics out of fairy tales), believed that if the state redivided income each morning, by night the rich would again be sleeping in their comfortable beds and the poor under the bridges. (I think she thought this a cogent argument against equilitarian taxes.)

Now, Paul Samuelson, aged 20 a hundred years later, was not Harriet Martineau or even David Ricardo; but as an equilibrium theorist he naturally tended to think of models in which things settle down to a unique position independently of initial conditions. Technically speaking, we theorists hoped not to introduce hystersis phenomena into our model, as the Bible does when it says, 'We pass this way only once' and, in so saying, takes the subject out of the realm of science into the realm of genuine history." -- Paul A. Samuelson, "What Classical and Neo-Classical Monetary Theory Really Was," Canadian Journal of Economics, V 1., N. 1 (1968): 1-15. (Reprinted in Monetary Theory: Selected Readings (edited by Robert W. Clower), Penguin (1969))